2.8Summary of Maxwell's Differential Laws and Integral Theorems

In this chapter, the divergence and curl operators have been
introduced. A third, the gradient, is naturally defined where it is
put to use, in Chap. 4. A summary of these operators in the three
standard coordinate systems is given in Table I at the end of the
text. The problems for Secs. 2.1 and 2.4 outline the
derivations of the gradient and curl operators in cylindrical and
spherical coordinates.

The integral theorems of Gauss and Stokes are two of three
theorems summarized in Table II at the end of the text. Gauss'
theorem states how the volume integral of any scalar that can be
represented as the divergence of a vector can be reduced to an
integration of the normal component of that vector over the surface
enclosing that volume. A volume integration is reduced to a surface
integration. Similarly, Stokes' theorem reduces the surface
integration of any vector that can be represented as the curl of
another vector to a contour integration of that second vector. A
surface integral is reduced to a contour integral.

These generally useful theorems are the basis for moving from the
integral law point of view of Chap. 1 to a differential point of view.
This transition from a global to a point-wise view of fields is
summarized by the shift from the integral laws of Table 1.8.1 to the
differential laws of Table 2.8.1.

The aspects of a vector field encapsulated in the divergence and
curl can always be recalled by returning to the fundamental
definitions, (2.1.2) and (2.4.2), respectively. The divergence is
indeed defined to represent the net outward flux through a closed
surface. But keep in mind that the surface is incremental, and that
the divergence describes only the neighborhood of a given point.
Similarly, the curl represents the circulation around an incremental
contour, not around one that is of finite size.

What should be committed to memory from this chapter? The
theorems of Gauss and Stokes are the key to relating the integral and
differential forms of Maxwell's equations. Thus, with these theorems
and the integral laws in mind, it is easy to remember the
differential laws. Applied to differential
volumes and surfaces, the theorems also provide the definitions (and
hence the significances) of the divergence and curl operators
independent of the coordinate system. Also, the evaluation in
Cartesian coordinates of these operators should be remembered.